Questions tagged [galois-cohomology]
The galois-cohomology tag has no usage guidance.
276
questions
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When is $B^G\backslash(B/A)^G$ finite?
Let $G$ be a locally compact group, let $A,B$ be (not necessarily abelian) connected reductive complex groups equipped with continuous actions of $G$ via algebraic automorphisms. Let $\phi:A\to B$ be ...
4
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0
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84
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Anisotropic semisimple groups with no real compact factor
Let $F$ be a number field, and let $G$ be a semi-simple connected, anisotropic algebraic group over $F$ which is $F$-simple (or almost simple, the question is agnostic to isogenies). Suppose further ...
2
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1
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100
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Connecting homomorphism in non-abelian cohomology
Let $G$ be a simply connected, semisimple algebraic group over $\mathbb{R}$ and let $X$ be a homogeneous space for $G$ with finite commutative stabilizer $\mu$. There is a connecting homomorphism from ...
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138
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cokernel of $H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/\operatorname{im}(\kappa_v)$
Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.
Let $\...
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Bounding dimension of $H^1(G_{\mathbb{Q}}, (V_pE)^{\otimes n})$
Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime of good reduction, $T_pE$ is its $p$-adic Tate module, $V_pE = T_pE\otimes \mathbb{Q}_p$, and $(V_pE)^{\otimes n}$ its $n$'th tensor ...
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315
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Amitsur's theorem for generalized Severi–Brauer varieties
Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...
4
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139
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A computation of nearby cycles
I'm currently reading P.Scholze's paper "THE LANGLANDS-KOTTWITZ APPROACH FOR THE MODULAR
CURVE". In Lemma 7.7, he showed a (maybe simple) nearby cycle computation, which I can't follow.
Now ...
3
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1
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221
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Global duality theorem for 2-part
$\DeclareMathOperator\coker{coker}\DeclareMathOperator\Sha{Sha}$Let $K$ be a number field.
Let $E/K$ be an elliptic curve over $K$.
Suppose finiteness of $\Sha(E/K)$.
According to Global duality ...
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72
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Inflation-restrction sequence for maximal $S$-ramified extension
Let $K$ be a number field. Let $G_K$ be an absolute Galois group of $K$. Let $M$ be a $G_K$-module and $L/K$ be a finite extension.
There is a inflation-restriction exact sequence,
$0\to H^1(Gak(L/K), ...
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Absolute Galois cohomology of function fields (of high-dimensional) varieties
What is known about the absolute Galois cohomology of function fields of varieties of dimension 2 or larger? Specifically, I am interested in multiplicative coefficients $\mathbb G_m$.
I have seen ...
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240
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Group homology for a metacyclic group
Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
We work with the first homology group
$$ H_1(G,M).$$
For any ...
8
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582
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$\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{R})$
Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an ...
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For which subgroups the transfer map kills a given element of a group?
$\newcommand{\ab}{{\rm ab}}
\newcommand{\ord}{{\rm ord}}
$Let $G$ be a finite or profinite group. Consider the abelianized group
$$G^\ab=G/G'$$
where $G'$ is the commutator subgroup of $G$.
Let $H\...
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Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$
This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763.
It got upvotes, but no answers or comments, and so I ask it here.
Let $G$ ...
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305
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Pontryagin dual of cokernel, $(\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $
Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.
Consider the natural map
$$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{...
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121
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Representability of twists of projective schemes
Let $K$ be a perfect field, and let $S$ be a projective $K$-scheme. Denote by $\text{Twist}(S/K)$ the set of twists of $S$ up to $K$-isomorphism. These are (apriori) sheaves $\mathcal{F}$ on the ...
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Possible questions about the Tate-Shafarevich subgroup of a Galois hypercohomology group?
$\newcommand{\wt}{\widetilde}$
Let $n=1,2$. There are infinite torsion abelian groups $H^1$, $H^2$ killed by some natural number $m$.
There are finite subgroups
$$ {\rm Sha}^1 \subset H^1,\quad ...
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120
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Kernel of restriction map in Galois cohomology
Let $S$ be the algebraic group $SL_2/\mathbb{Q}_p$ with a $G=G_{\mathbb{Q}}$ action, (acts by conjugation with a representation $\rho: G\longrightarrow GL_2$.)
Let $G_p$ be the decomposition group at ...
1
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1
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173
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Crystalline fibre of a morphism of Galois cohomology stacks
Let $K = \mathbb{Q}_p$, $G = G_K$ its absolute Galois group. Let
$$1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$$
be a split exact sequence of (not necessarily abelian) group ...
2
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1
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275
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Equivalence between twists of a curve and torsors of its automorphism group
Let $X$ be a curve defined over a number field $K$, and let $G_K$ be the absolute Galois group of $K$. Let $\text{Aut}(X)$ be the group of $\overline{K}$-defined automorphisms of $X$, and consider the ...
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80
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Algebraizable image of a morphism of Galois cohomology stacks
Assume I have a surjective morphism of algebraic group schemes over $\mathbb{Q}_p$, $\mathcal{G}\longrightarrow S$, equipped with a section, and assume both of these group schemes are equipped with an ...
6
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2
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358
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Twisted forms with real points of a real Grassmannian
Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$.
We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{\...
2
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0
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100
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Extensions of groups with a $G$-action
Let $1\longrightarrow A\longrightarrow \mathcal{G}\longrightarrow R\longrightarrow 1$ be an exact sequence of algebraic group schemes, with $\mathcal{G}$ being an extension of $R$, an affine reductive ...
3
votes
1
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223
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The second Tate-Shafarevich group of a permutation module is trivial
Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $...
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220
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If $H$ is a quotient of $G$, does there exist an $H$-extension of $\mathbb{Q}$ not contained in a $G$-extension?
Let $\phi\colon G\rightarrow H$ be a surjective homomorphism between finite groups. Assume that $\phi$ is not split, in other words there exists no homomorphism $\sigma\colon H\rightarrow G$ such that ...
6
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336
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Ker of corestriction of Galois cohomology
Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.
Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.
On the other hand, ...
3
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123
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Cohomology of local fields in positive characteristic
It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...
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2
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520
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Example of continuous cohomology vs cohomology
I am looking for an example of a locally compact group $G$ and a continuous $G$ module $M$, which also is locally compact, such that the continuous cochain cohomology differs from group cohomology (...
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142
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About the exact sequence $0\to $$Ш(E/\Bbb{Q})[\phi]\to Ш(E/\Bbb{Q})[2]\to Ш(E'/\Bbb{Q})[\hat{\phi}]$ in Silverman's book of elliptic curves
This question is about Silverman's book,$7$-th line from the bottom of $350$ page of 'The arithmetic of elliptic curves' (http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf) .
Let $E$ ...
2
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1
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333
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Galois cohomology of Tate modules
Let $E,E'$ be a pair of elliptic curves defined over $\mathbb{Z}$. Let $T_p[E], T_p[E']$ be their associated ($p$-adic) Tate modules. These are Galois representations for the absolute Galois group of $...
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0
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167
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Crystalline exact sequence in Galois cohomology
Let $G$ be the absolute Galois group of $\mathbb{Q}_p$, and let $1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$ be a short exact sequence of (non-abelian) algebraic group ...
3
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1
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209
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Deformations of Galois cohomology
Let $M = (\mathbb{Z}_p)^2$ be a Galois representation, with Galois action given by $\rho: G\longrightarrow SL_2(\mathbb{Z}_p)$. I am trying to understand how sensitive the Galois cohomology group $H^1(...
3
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1
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335
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Local Tate duality for F-vector space
A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q_p$, $A$ be a finite $G_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect ...
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Biquadratic extension of global function fields with cyclic decomposition groups
Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$.
Question. What would be an example of a ...
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1
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137
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Decomposition groups for the Galois module $\mu_8$
$\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Gal}{Gal}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Fbar}{{\overline F}}
\newcommand{\G}{\...
2
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292
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Galois cohomology of $\breve{\mathbb Q}_p \otimes_{\mathbb Q_p} \breve{\mathbb Q}_p$
Let $\breve{\mathbb Q}_p$ denote the completion of the maximal unramified extension of $\mathbb Q_p$. I‘d like to compute Galois cohomology groups and sets related to $\breve{\mathbb Q}_p \otimes_{\...
5
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241
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Torus gerbes over curves
Setup: Let $k$ be an algebraically closed field. Let $C$ be a smooth connected curve over $k$. Let $K(C)$ be the function field of $C$.
Tsen's Theorem implies that every $\mathbb{G}_m$-gerbe over $K(C)...
4
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1
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119
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Real forms of the general linear Lie superalgebra
I'm interested in a classification of the real forms of the general linear Lie superalgebra $\mathfrak{gl}_{m|m}(\mathbb{C})$.
The real forms of the simple complex Lie superalgebras were classified by ...
4
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Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$
$\newcommand{\GL}{\operatorname{GL}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\C}{{\Bbb C}}
$For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle
of $G=\GL_{n,\R}\,$, that is,
an invertible ...
5
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299
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Do algebraic tori have no $H^1$?
If $G$ is an algebraic group over a field $K$, we can consider the Galois (or flat) cohomology $H^1(K, G)$. If $G = \mathbb{G}_a$ or $\mathbb{G}_m$, it is well known that $H^1(K, G) = 0$ (the latter ...
3
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1
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161
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Why an isogeny induces a surjection between points over maximal unramified extension?
Let $E$ and $E'$ be elliptic curves over $\mathbb Q$, and let $\phi\colon E\to E'$ be an isogeny defined over $\mathbb Q$. Let $p$ be a prime relatively prime to the degree of $\phi$. Let $\mathbb Q_p^...
3
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Finiteness for Galois cohomology for $\mathbb{Z}_p$-module coefficients
I am looking for a general survey on the finite generation properties of
$$H^i(F,\mathbb{Z}_p(j))$$
for fields $F$. Here I refer to Galois cohomology (continuous group cohomology) and the group is ...
2
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0
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102
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Galois cohomology with coefficients in the integers of the Lubin-Tate extension
Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...
2
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1
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137
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Vanishing of the degree 2 cohomology of a p-adic field with coefficients Q/Z and action of the Frobenius and the Pontryagin dual of the inertia
Let $K$ be a $p$-adic field with Galois group $G$ and inertia subgroup $I\subset G$. Denote $(-)^\ast=\mathrm{Hom}_{cont}(-,\mathbb{Q}/\mathbb{Z})$. Using Tate local duality, we can compute $$H^2(G,\...
5
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1
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169
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Restriction vs. multiplication by $n$ in Tate cohomology
$\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Cor}{Cor}$
This question was asked in MSE.
It got no answers or comments, and so I post it here.
Let $H$ be a subgroup of a finite group $G$, and ...
4
votes
1
answer
293
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Why descend a representation (of a finite group) over $K$ to a representation over $k$ with $k$ a subfield of $K$ is useful?
I heard that Schur was trying to answer the following question
Given a representation of a finite group $G \overset{\rho}{\rightarrow} \operatorname{GL}_{n}(K)$, how to find the smallest subfield $k$ ...
7
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2
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846
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Is this exact sequence known?
$\newcommand{\Tors}{{\rm Tors}}
\newcommand{\tf}{{\rm\, t.f.}}
\newcommand{\Gt}{{\Gamma\!,\,\Tors}}
\newcommand{\Gtf}{{\Gamma\!,\tf}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\...
4
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1
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606
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Generalizations of global Euler characteristic formula
Let $ K $ be a number field, $ S $ a finite set of primes of $K $ including the archimedean primes and $ G_{K,S} $ be the Galois group of the maximal extension of $K$ unramified outside $ S $. Assume ...
2
votes
1
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175
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Image of Kummer map for CM Elliptic curves
Let $K$ be an imaginary quadratic field and let $F$ be a finite extension of $K$. Let $E$ be an elliptic curve over $F$ with CM by $K$. Suppose that $p$ is a prime that splits as $p=\pi\pi^*$ in $K$. ...
5
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0
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199
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Essence of relations between central simple algebras and Galois cohomology in canonical morphism of class field theory
I was somewhat puzzled after I finished learning class field theory for several times. My question is about the relations between "classical simple algebras, Brauer groups" and "modern ...